Question

The value in the table represent an exponential function. What is the common ratio of associated geometric sequence?

x y
1 9
2 36
3 114
4 576
5 2304


45
9
27
4

Answer 1

In an exponential function, the relationship between \(y\) and \(x\) can often be expressed in the form \(y = a \cdot b^x\), where \(a\) is the initial value, and \(b\) is the base of the exponential function.

To find the common ratio of the associated geometric sequence of the values of \(y\) in the given table, we will compute the ratios between consecutive \(y\) values.

Here are the \(y\) values from the table:

- \(y_1 = 9\)
- \(y_2 = 36\)
- \(y_3 = 114\)
- \(y_4 = 576\)
- \(y_5 = 2304\)

Let's calculate the ratios:

1. Ratio of \(y_2\) to \(y_1\):
\[
\frac{y_2}{y_1} = \frac{36}{9} = 4
\]

2. Ratio of \(y_3\) to \(y_2\):
\[
\frac{y_3}{y_2} = \frac{114}{36} = 3.1667 \quad (\text{approximately})
\]

3. Ratio of \(y_4\) to \(y_3\):
\[
\frac{y_4}{y_3} = \frac{576}{114} = 5.0526 \quad (\text{approximately})
\]

4. Ratio of \(y_5\) to \(y_4\):
\[
\frac{y_5}{y_4} = \frac{2304}{576} = 4
\]

From the ratios we've calculated, it is clear they are not constant, which indicates that this is not a simple geometric sequence with a constant common ratio.

However, the ratio of the first and last calculations both yield 4, which indicates two parts of the sequence have this relationship.

Thus, while not consistent throughout, some parts yield a common ratio of \(4\).

Therefore, based on our calculations, the most consistent answer is:

**4**